#263796

order	512
nilpotency class	3
isoclinism	isoclinic class of #263779
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3561	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,12)(5,8)(6,10)(7,14)(9,15)(13,16)(17,18,22,26)(19,27,30,41)(20,31,33,46)(21,34,35,25)(23,36,37,29)(24,39,40,32)(28,43,45,38)(42,47,48,44);(1,3,5,12)(2,4,8,11)(6,7,15,13)(9,14,10,16)(17,19,21,29,22,30,35,36)(18,23,25,27,26,37,34,41)(20,28,32,44,33,45,39,47)(24,38,31,42,40,43,46,48);(1,4,5,11)(2,7,8,13)(3,10,12,9)(6,16,15,14)(23,37)(27,41)(38,43)(42,48);(2,8)(4,11)(9,10)(14,16)(17,20,21,32,22,33,35,39)(18,24,25,31,26,40,34,46)(19,28,29,44,30,45,36,47)(23,38,27,42,37,43,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170992);